Atoms in the p-localization of Stable Homotopy Category

We study p-localizations, where p is an odd prime, of the full subcategories \( {\mathcal{S}}^n \) of stable homotopy category formed by CW-complexes with cells in n successive dimensions. Using the technique of triangulated categories and matrix problems, we classify the atoms (indecomposable objects) in \( {\mathcal{S}}_p^n \) for n ≤ 4(p − 1) and show that, for n > 4(p − 1), this classification is wild in a sense of the representation theory.